With many thanks to Crayz9000, Arthur_Tuxedo, phongn, Stark, and GrandMasterTerwynn for their advice, my upgraded system is now operational. I'm still waiting to see if I can snag a Radeon 9500 Pro for a decent price on eBay, but the CPU, RAM, and mobo upgrades are done and working.
Asus A7N8X Deluxe motherboard
AMD Athlon XP "Barton" 2800+ CPU (Newegg had a special on these when I ordered; they were just a little more than the 2500+, which Crayz9000 had recommended.)
512 MB PC2700 RAM
I didn't expect the stock fan/heatsink that came with the CPU to be so noisy (the one for my old P3-866 was nearly silent -- this new one is akin to a heavy-duty electric razor), but otherwise, the entire process has gone off without any problems at all. Again, thanks to those that helped me out here.
Upgraded system: online
Moderator: Thanas
Upgraded system: online
Member of the Anti-PETA Anti-Fascist League